Decoding apparatus, decoding method, and decoding program for use in quantum error correction

ABSTRACT

According to one embodiment, a decoding apparatus includes first and second acquisition units, a holding unit, a calculation unit, and a decision unit. The first acquisition unit acquires first measurement values of measurements performed to measure an eigenvalue of an encoded Z operator to a first encoded qubit of the two encoded qubits. The second acquisition unit acquires second measurement values of measurements performed to measure an eigenvalue of an encoded X operator to a second encoded qubit of the two encoded qubits. The holding unit holds error probabilities for the first measurement values and the second measurement values. The calculation unit calculates probabilities for measurement values of an encoded Bell measurement by using the first measurement values, the second measurement values, and the error probabilities. The decision unit decides measurement values of the encoded Bell measurement, based on the calculated probabilities.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based upon and claims the benefit of priority fromJapanese Patent Application No. 2012-239558, filed Oct. 30, 2012, theentire contents of which are incorporated herein by reference.

FIELD

Embodiments described herein relate generally to a decoding apparatus,decoding method, and decoding program for use in quantum errorcorrection and fault-tolerant quantum computation.

BACKGROUND

Since a quantum computer uses the state of quantum mechanicalsuperposition, decoherence by which this state breaks causes a memoryerror or gate error. This is a problem that is unique to the quantumcomputer and does not arise in the conventional classical computers.Therefore, quantum error correction capable of correcting an error likethis and fault-tolerant quantum computation that performs reliablequantum computation by using the quantum error correction are regardedas indispensable in the quantum computer.

Theoretically, reliable quantum computation can be executed as long aspossible if the error probability is lower than a certain threshold (thethreshold theorem). The threshold depends on the method offault-tolerant quantum computation, and the present highest value isstill low, i.e., about 1%. In addition, even when the threshold is 1%,the resources (the qubit count and (or) the gate count) become enormous.Therefore, demands have arisen for a better fault-tolerant quantumcomputation method.

Classical error correction has dramatically improved the performance bychanging algebraic hard-decision decoding to soft-decision decodingbased on probabilistic inference. Accordingly, it may be possible toimprove the performance of fault-tolerant quantum computation bysoft-decision decoding. Note that there is a related art of quantumerror correction using soft-decision decoding based on probabilisticinference.

Unfortunately, fault-tolerant quantum computation using soft-decisiondecoding has not been studied yet, and its performance is unknown.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

FIG. 1 is a view showing error-correcting teleportation;

FIG. 2 is a view showing a decoding apparatus according to the firstembodiment;

FIG. 3 is a view for explaining transversality as the feature of Z·Xseparation type stabilizer codes;

FIG. 4 is a flowchart of a subroutine performed by a probabilitycalculation unit of the decoding apparatus according to the firstembodiment;

FIG. 5 is a flowchart indicating an operation including FIG. 4 of theprobability calculation unit of the decoding apparatus according to thefirst embodiment;

FIG. 6 is a flowchart of subroutine 1 of the probability calculationunit of the decoding apparatus according to the first embodiment;

FIG. 7 is a flowchart of subroutine 2 including FIG. 6 of theprobability calculation unit of the decoding apparatus according to thefirst embodiment;

FIG. 8 is a flowchart indicating an operation including FIGS. 5 and 7 ofthe probability calculation unit of the decoding apparatus according tothe first embodiment;

FIG. 9 is a view showing a decoding apparatus according to the secondembodiment;

FIG. 10 is a view showing a decoding apparatus according to the thirdembodiment;

FIG. 11 is a view showing the performance of the decoding apparatusaccording to the third embodiment;

FIG. 12 is a view showing a decoding apparatus according to the fourthembodiment;

FIG. 13 is a view showing an encoding controlled-NOT gate including adecoding apparatus according to the fifth embodiment;

FIG. 14 is a graph showing the simulation results of the first example;

FIG. 15 is a graph showing the simulation results of the third example;and

FIG. 16 is a graph showing the simulation results of the fourth example.

DETAILED DESCRIPTION

Decoding apparatuses, decoding methods, and decoding programs accordingto embodiments will be explained in detail below with reference to theaccompanying drawing. Note that in the following embodiments, partsdenoted by the same reference numbers perform the same operations, and arepetitive explanation will be omitted.

The embodiments have been made in consideration of the above situations,and provide a decoding apparatus, decoding method, and decoding programfor performing soft decision that improves the performance offault-tolerant quantum computation.

According to one embodiment, a decoding apparatus for use in encodedBell measurement for two encoded qubits, includes a first acquisitionunit, a second acquisition unit, a holding unit, a calculation unit, anda decision unit. The first acquisition unit acquires first measurementvalues of measurements performed to measure an eigenvalue of an encodedZ operator with respect to a first encoded qubit of the two encodedqubits. The second acquisition unit acquires second measurement valuesof measurements performed to measure an eigenvalue of an encoded Xoperator with respect to a second encoded qubit of the two encodedqubits. The holding unit holds error probabilities for the firstmeasurement values and the second measurement values. The calculationunit calculates probabilities for measurement values of the encoded Bellmeasurement by using the first measurement values, the secondmeasurement values, and the error probabilities. The decision unitdecides a measurement value of the encoded Bell measurement, based onthe probabilities calculated by the calculation unit.

When compared to the conventional methods, the decoding apparatuses,decoding methods, and decoding programs of the embodiments canremarkably improve the decoding performance of encoded Bell measurementfor use in error-correcting teleportation or an encoding controlled-NOTgate. As a result, the performance of fault-tolerant quantum computationcan be improved.

(First Embodiment)

This embodiment focuses attention on error-correcting teleportation as aquantum error correction method suitable for fault-tolerant quantumcomputation. FIG. 1 shows the operation of error-correctingteleportation (see E. Knill, Nature 434, 39 [2005]).

In error-correcting teleportation, so-called Bell measurement isperformed between an input state ((1-1)) as an error correction targetand a first qubit in a Bell state ((1-2)). In the following description,((E1-E2)) means an expression or symbol indicated by (E1-E2). Each of E1and E2 represents an integer of 1 or more. For example, “an input state((1-1))” means “an input state |ψ_(in)>_(L)”.|ψ_(in)

_(L)  (1-1)|0

_(L)|0

_(L)+|1

_(L)|1

_(L)  (1-2)Each qubit is encoded into a given quantum error correction code inadvance (a suffix L represents this). Therefore, this Bell measurementwill be called “encoded Bell measurement” hereinafter. Since each bit isencoded beforehand, it is possible to obtain highly reliable measurementresults by encoded Bell measurement. An encoded X gate and encoded Zgate are executed on a second qubit in the Bell state in accordance withthe measurement results of this encoded Bell measurement. The secondqubit ((2-1)) in the Bell state thus obtained is in a state in which anerror of the input state ((1-1)) is corrected. This is error-correctingteleportation.|ψ_(out)

_(L)  (2-1)

Encoded Bell measurement normally includes an encoding controlled-NOTgate, and subsequent eigenvalue measurements of an encoded Z operatorand encoded X operator. Conventionally, hard-decision decoding isindependently performed on the measurement results of two encodedqubits. By contrast, a decoding apparatus of this embodimentsimultaneously processes the measurement results of two encoded qubits,and estimates and determines measurement results by soft-decisiondecoding based on probabilistic inference.

Next, a decoding apparatus according to the first embodiment will beexplained with reference to FIG. 2. FIG. 2 shows the arrangement of thedecoding apparatus according to the first embodiment.

The decoding apparatus of this embodiment includes an M_(z) measurementvalue input unit 201, M_(x) measurement value input unit 202, errorprobability holding unit 203, probability calculation unit 204, anddecision unit 205.

Note that all decoding apparatuses of embodiments are applicable to alltypes of quantum computers regardless of how physical qubits areimplemented. (Examples of the physical qubits are the polarization/spacemodes of a photon, the energy levels, electron spins, and nuclear spinsof cooled ions or neutral atoms, electron spins and nuclear spins in asolid, superconducting Josephson qubits, and the energy levels andelectron spins of semiconductor quantum dots).

M_(z) and M_(x) respectively represent units of physical qubitmeasurements performed for the eigenvalue measurements of the encoded Zoperator and encoded X operator necessary for encoded Bell measurement.The M_(z) measurement value input unit 201 and M_(x) measurement valueinput unit 202 respectively receive the measurement values of M_(z) andM_(x) and output them to the probability calculation unit 204.

The error probability holding unit 203 holds the error probabilities forthe measurement values of M_(z) and M_(x) in advance, or holds anexternally input error probabilities. The error probability holding unit203 outputs this error probabilities to the probability calculation unit204. Note that this error probabilities can be updated as needed.

By using the measurement values of M_(z) and M_(x) respectively inputfrom the M_(z) measurement value input unit 201 and M_(x) measurementvalue input unit 202, the error probabilities for the measurement valuesof Ms and M_(x), which are input from the error probability holding unit203, and information of the quantum error correction code, theprobability calculation unit 204 calculates the probabilities for themeasurement values of Bell measurement, i.e., calculates theprobabilities for the eigenvalues of the encoded Z operator with respectto the first qubit in the Bell state, and the probabilities for theeigenvalues of the encoded X operator with respect to the input state,and outputs the calculated probabilities to the decision unit 205. Notethat the quantum error correction code information is held in advance orexternally input and held.

Based on the probabilities input from the probability calculation unit204, the decision unit 205 decides the measurement values of Bellmeasurement, i.e., the measurement value of the eigenvalue of theencoded Z operator with respect to the first qubit in the Bell state,and the measurement value of the eigenvalue of the encoded X operatorwith respect to the input state. Then, the decision unit 205 outputs themeasurement values and terminates the decoding process.

A standard method of the above-mentioned decision is a method ofdetecting a maximum one of the probabilities input from the probabilitycalculation unit 204, and decides that the corresponding eigenvalues arethe measurement values.

The operation of the probability calculation unit 204 of the firstembodiment will be explained in more detail below. (The operation of theprobability calculation unit 204 herein explained similarly applies tothe second embodiment, and almost similarly applies to a probabilitycalculation unit 1004 of the third and fourth embodiments to bedescribed later.)

Assume that the quantum error correction code is a stabilizer code (seeM. A. Nielsen and I. L. Chuang, Quantum Information and Computation,Cambridge Univ. Press [2000], Chapter 10, pp. 425-499). For the sake ofsimplicity, the explanation will be made by assuming that one qubit isencoded by n qubits. However, the same operation applies even when twoor more qubits are encoded.

Assume that each stabilizer generator includes only a Z operator andidentity operator, or only an X operator and identity operator, andthese stabilizer generators will be called a Z stabilizer generator andX stabilizer generator. A stabilizer code like this will be called “aZ·X separation type stabilizer code”. (This code is also called a CSScode, and a typical example is a Steane's 7-qubit code. See M. A.Nielsen and I. L. Chuang, Quantum Information and Computation, CambridgeUniv. Press [2000], Chapter 10, pp. 425-499). Letting n_(z) be thenumber of Z stabilizer generators and n_(x) be that of X stabilizergenerators, n=n_(z)+n_(x)+1 holds. Assume also that an encoded Zoperator Z_(L) includes only the Z operator and identity operator, andan encoded X operator X_(L) includes only the X operator and identityoperator.

An important property of the Z·X separation type stabilizer code is theability to transversally execute the encoded controlled-NOT gate and themeasurements of the eigenvalues of the encoded Z operator and encoded Xoperator necessary for encoded Bell measurement (see M. A. Nielsen andI. L. Chuang, Quantum Information and Computation, Cambridge Univ. Press[2000], Chapter 10, pp. 425-499). That is, to execute the encodedcontrolled-NOT gate on two encoded qubits, the physical controlled-NOTgate need only be executed on the j^(th) physical qubit (j is an integerof 1 to n) of each encoded qubit. Also, to measure the eigenvalue of theencoded Z operator with respect to a given encoded qubit, the eigenvalueof the Z operator need only be measured for the physical qubit of theencoded qubit. This applies to eigenvalue measurement of the encoded Xoperator. FIG. 3 shows the foregoing (when n=3). An M_(z) measurementvalue input unit 301 receives three M_(z) measurement values, and anM_(x) measurement value input unit 302 receives three M_(x) measurementvalues.

In the following explanation, the stabilizer generator, encoded Zoperator Z_(L), and encoded X operator X_(L) are each represented by amatrix containing 0 and 1 as components. When n=4, for example, Zstabilizer generator S_(z)=ZIIZ is represented by matrix S_(z)=(1 0 0 1)(I is an identity operator). Similarly, X stabilizer generatorS_(x)=XIIX is represented by S_(x)=(1 0 0 1).

Also, measurement values ((3-1)) input to the M_(z) measurement valueinput unit 201, measurement values ((3-2)) input to the M_(x)measurement value input unit 202, and errors ((4-1)) and ((4-2)) forthese measurement values are each represented by a matrix (or vector)containing 0 and 1 as components.

$\begin{matrix}\{ {{ m_{zj} \middle| j  = 1},2,\ldots\mspace{14mu},n} \} & ( {3\text{-}1} ) \\\{ {{ m_{xj} \middle| j  = 1},2,\ldots\mspace{14mu},n} \} & ( {3\text{-}2} ) \\\{ {{ e_{zj} \middle| j  = 1},2,\ldots\mspace{14mu},n} \} & ( {4\text{-}1} ) \\\{ {{ e_{xj} \middle| j  = 1},2,\ldots\mspace{14mu},n} \} & ( {4\text{-}2} )\end{matrix}$For example, if ((5-1)) holds for n=4, this is represented by ((5-2)).

$\begin{matrix}{{m_{z\; 1} = 1},{m_{z\; 2} = {- 1}},{m_{z\; 3} = {- 1}},{m_{z\; 4} = 1}} & ( {5\text{-}1} ) \\{\overset{arrow}{m_{z}} = \begin{pmatrix}0 \\1 \\1 \\0\end{pmatrix}} & ( {5\text{-}2} )\end{matrix}$Likewise, ((6-1)) is represented by ((6-2)).

$\begin{matrix}{{m_{x\; 1} = 1},{m_{x\; 2} = {- 1}},{m_{x\; 3} = {- 1}},{m_{x\; 4} = 1}} & ( {6\text{-}1} ) \\{\overset{arrow}{m_{x}} = \begin{pmatrix}0 \\1 \\1 \\0\end{pmatrix}} & ( {6\text{-}2} )\end{matrix}$

The probability calculation unit 204 calculates the probabilities((7-1)) for the measurement values of encoded Bell measurement, i.e.,the probabilities ((7-1)) for eigenvalues m_(z) of the encoded Zoperator Z_(L) with respect to the first qubit in the Bell state andeigenvalues m_(x) of the encoded X operator X_(L) with respect to theinput state, by using the measurement values ((7-2)) input from theM_(z) measurement value input unit 201 to the probability calculationunit 204, the measurement values ((7-3)) input from the M_(x)measurement value input unit 202 to the probability calculation unit204, and the error probabilities ((7-4)) input from the errorprobability holding unit 203 to the probability calculation unit 204.P(m _(z) ,m _(x))  (7-1){right arrow over (m _(z))}  (7-2){right arrow over (m _(x))}  (7-3)P ⁽⁰⁾({right arrow over (e _(z))},{right arrow over (e _(x))})  (7-4)

Because the above-described transversality exists, it is in many casespossible to represent the error probabilities ((7-4)) by the product ofthe error probabilities ((9-1)) (j is an integer of 1 to n) of n bitpairs, as indicated by equation (8-1) below.

$\begin{matrix}{{P^{(0)}( {\overset{arrow}{e_{z}},\overset{arrow}{e_{x}}} )} = {\prod\limits_{j = 1}^{n}{P_{j}^{(0)}( {e_{zj},e_{xj}} )}}} & ( {8\text{-}1} ) \\{P_{j}^{(0)}( {e_{zj},e_{xj}} )} & ( {9\text{-}1} )\end{matrix}$It should be noted that an error e_(zj) for m_(zj) and an error e_(xj)for m_(xj) are correlated because of the physical controlled-NOT gate,and this correlation is taken into consideration by the errorprobability ((9-1)).

First, the relative probabilities ((11-1)) are calculated by equation(10-1) below.

$\begin{matrix}{{R( {m_{z},m_{x}} )} = {\sum\limits_{({\overset{arrow}{l_{z}},\overset{arrow}{l_{x}}})}{P^{(0)}( {{\overset{arrow}{l_{z}} + \overset{arrow}{m_{z}}},{\overset{arrow}{l_{x}} + \overset{arrow}{m_{x}}}} )}}} & ( {10\text{-}1} ) \\{R( {m_{z},m_{x}} )} & ( {11\text{-}1} )\end{matrix}$The sum ((12-1)) is taken for ((12-2)) that satisfies conditions 1 to 4below.

$\begin{matrix}\sum\limits_{({\overset{arrow}{l_{z}},\overset{arrow}{l_{x}}})} & ( {12\text{-}1} ) \\( {\overset{arrow}{l_{z}},\overset{arrow}{l_{x}}} ) & ( {12\text{-}2} )\end{matrix}$

-   Condition 1: ((13-1)) holds for all the Z stabilizer generators    S_(z).-   Condition 2: ((13-2)) holds for all the X stabilizer generators    S_(x).-   Condition 3: ((13-3)) holds for the encoded Z operator Z_(L).-   Condition 4: ((13-4)) holds for the encoded X operator X_(L).    S _(Z){right arrow over (l _(z))}=0  (13-1)    S _(X){right arrow over (l_(x))}=0  (13-2)    Z _(L){right arrow over (l_(z))}=m _(z)  (13-3)    X _(L){right arrow over (l_(x))}=m _(x)  (13-4)    The above matrix operation is a normal binary operation. (The    eigenvalue m_(x) of the encoded Z operator Z_(L) and the eigenvalue    m_(x) of the encoded X operator X_(L) are also binary values. That    is, an eigenvalue of 1 is represented by a binary value of 0, and an    eigenvalue of −1 is represented by a binary value of 1.).

((12-2)) has a total of 2n variables, and they have 2^(2n) patterns.Since, however, the total number of above-mentioned conditions is(n_(z)+n_(x)+1+1)=(n+1) (when using relation n=n_(z)+n_(x)+1), ((12-1))can independently take (n−1) variables, and they have 2^(n−1) patternsfewer than 2^(2n) patterns.

By taking account of this in the calculation of the probabilitycalculation unit 204, ((12-1)) is executed by moving (n−1) variables of((12-2)) as independent variables, and determining (n+1) remainingvariables in accordance with conditions 1 to 4 described above.

The above-mentioned subroutine executed by the probability calculationunit 204 will be explained below with reference to FIG. 4. FIG. 4 is aflowchart of the subroutine performed by the probability calculationunit of the decoding apparatus according to the first embodiment whenusing the Z·X separation type stabilizer code.

Initial values R(m_(z),m_(x)) (i=0) for taking the sum of the relativeprobabilities ((10-1)) are set to zero (step S401). i takes 2^(n−1)values as the number of patterns of the variables which ((12-1)) canindependently take.

i is increased by 1 (step S402).

Whether i is 2^(n−1) or less is determined. If i is 2^(n−1) or less, theprocess advances to step S404. If i is not 2^(n−1) or less, it isdetermined that the relative probability calculation is complete (stepS403).

Of all components l_(z)[j] and l_(x)[j](j=1, . . . , n) of two vectorscontained ((12-2)), each bit of i is substituted into (n−1) components(step S404).

Of all the components l_(z)[j] and l_(x)[j], (n+1) remaining componentsare determined in accordance with above-mentioned conditions 1 to 4(step S405).

By using all the components l_(z)[j] and l_(x)[j] calculated in stepsS404 and S405, one term of the sum of the right-hand side of equation(10-1) above is calculated (step S406). The result is added to the valuecalculated for i last time, thereby finally obtaining the sum of theright-hand side of equation (10-1) above.

Steps S402 to S406 are repeated immediately before i exceeds 2^(n−1).

Finally, ((7-1)) is obtained by normalizing the relative probability((11-1)) as follows.

$\begin{matrix}{{P( {m_{z},m_{x}} )} = \frac{R( {m_{z},m_{x}} )}{\sum\limits_{({m_{z}^{\prime},m_{x}^{\prime}})}{R( {m_{z}^{\prime},m_{x}^{\prime}} )}}} & ( {14\text{-}1} )\end{matrix}$

The above operation of the probability calculation unit 204 will beexplained with reference to FIG. 5. FIG. 5 is a flowchart showing theoperation of the probability calculation unit of the decoding apparatusaccording to the first embodiment when using the Z·X separation typestabilizer code.

Since each of m_(z) and m_(x) takes a value of 0 or 1, summationperformed in the denominator of the right-hand side of equation (14-1)can take four patterns. Steps S501 to S503 are repeated for these fourpatterns.

In step S502, the subroutine shown in FIG. 4 is performed on the valuesof m_(z) and m_(x), thereby obtaining ((11-1)).

After the loop from step S501 to step S503 is complete, ((7-1)) isobtained by calculating the right-hand side of equation (14-1) inaccordance with ((11-1)) calculated by this loop.

The stabilizer code probability calculation described above becomesinefficient when the size of the code increases. Therefore, a practicalexample where efficient calculations are possible will be explained indetail below. (The operation of the probability calculation unit 204herein explained naturally similarly applies to the second embodiment,and almost similarly applies to a probability calculation unit 1004 ofthe third and fourth embodiments to be described later.)

Assume that a quantum error correction code is a concatenated codeformed by concatenating Z·X separation type stabilizer codes (see M. A.Nielsen and I. L. Chuang, Quantum Information and Computation, CambridgeUniv. Press [2000], Chapter 10, pp. 425-499). (Many such codes exist.Typical examples are a concatenated code of Steane's 7-qubit codes and aC₄/C₆ code to be described later. See E. Knill, Nature 434, 39 [2005].)A calculation herein explained is an efficient calculation when codes atindividual levels forming a concatenated code are small. For the sake ofsimplicity, a concatenated code obtained by concatenating, to level L,identical stabilizer codes for encoding one qubit by n qubits will beexplained. However, it is also possible to use stabilizer codes forencoding two or more qubits, or use different stabilizer codes atdifferent levels.

Assume that the error probability input from the error probabilityholding unit 203 to the probability calculation unit 204 is independentfor each block at level 1, and the error probability in a block b₁(b₁=1, 2, . . . , n^(L−1)) at level 1 is represented by ((15-1)).P _(b) ₁ ⁽⁰⁾({right arrow over (e _(z))},{right arrow over (e_(x))})  (15-1)It should be noted that the correlation between the error probability inthe block b₁ at level 1 of the first qubit in the Bell state and that ofthe block b₁ at level 1 in the input state is taken into consideration.

Because the above-described transversality exists, it is in many casespossible to represent the error probability ((15-1)) by the product ofthe error probabilities of n bit pairs forming the block b₁ at level 1.

The probability calculation unit 204 calculates the probabilities((16-1)) for the measurement values of encoded Bell measurement, i.e.,the probabilities ((16-1)) for the eigenvalues m_(z) of the encoded Zoperator at level L with respect to the first qubit in the Bell stateand the eigenvalues m_(x) of the encoded X operator at level L withrespect to the input state, by using measurement values ((7-2)) inputfrom the M_(z) measurement value input unit 201 to the probabilitycalculation unit 204, the measurement values ((7-3)) input from theM_(x) measurement value input unit 202 to the probability calculationunit 204, and the above-mentioned error probabilities ((15-1)) inputfrom the error probability holding unit 203 to the probabilitycalculation unit 204.P ^((L))(m _(z) ,m _(x))  (16-1)

First, the probabilities ((17-1)) for the eigenvalues m_(z) of theencoded Z operator at level 1 and the eigenvalues m_(x) of the encoded Xoperator at level 1 are calculated for the first qubit in the Bell stateand each block b₁ at level 1 in the input state by using theabove-described calculation method shown in FIGS. 4 and 5.P _(b) ₁ ⁽¹⁾(m _(z) ,m _(x))  (17-1)

Then, the probabilities ((18-1)) for the eigenvalues m₂ of the encoded Zoperator at level 2 and the eigenvalues m_(x) of the encoded X operatorat level 2 are calculated for the first qubit in the Bell state and eachblock b₂ (b₂=1, 2, . . . , n^(L−2)) at level 2 in the input state asfollows by using ((17-1)). b₁ and b₂ have the following relationship:b₁=n(b₂−1)+d (d=1, 2, . . . , n).P _(b) ₂ ⁽²⁾(m _(z) ,m _(x))  (18-1)

First, the relative probabilities ((19-2)) are calculated by equation(19-1) below.

$\begin{matrix}{{R_{b_{2}}^{(2)}( {m_{z},m_{x}} )} = {\sum\limits_{m_{z}^{(1)},m_{x}^{(1)}}\mspace{14mu}{\ldots\mspace{14mu}{\sum\limits_{m_{z}^{(n)},m_{x}^{(n)}}{\prod\limits_{d = 1}^{n}{P_{{n{({b_{2} - 1})}} + d}^{(1)}( {m_{z}^{(d)},m_{x}^{(d)}} )}}}}}} & ( {19\text{-}1} ) \\{\mspace{20mu}{R_{b_{2}}^{(2)}( {m_{z},m_{x}} )}} & ( {19\text{-}2} )\end{matrix}$In this case, the sum ((20-1)) is taken for ((20-2)) satisfyingconditions 5 to 8 below.

$\begin{matrix}{\sum\limits_{m_{z}^{(1)},m_{x}^{(1)}}\mspace{14mu}{\ldots\mspace{14mu}\sum\limits_{m_{z}^{(n)},m_{x}^{(n)}}}} & ( {20\text{-}1} ) \\{m_{z}^{(1)},m_{x}^{(1)},\ldots\mspace{14mu},m_{z}^{(n)},m_{x}^{(n)}} & ( {20\text{-}2} )\end{matrix}$

-   Condition 5: ((21-1)) holds for all Z stabilizer generators S_(z) at    level 2.-   Condition 6: ((21-2)) holds for all X stabilizer generators S_(X) at    level 2.-   Condition 7: ((21-3)) holds for the encoded Z operator Z_(L) at    level 2.-   Condition 8: ((21-4)) holds for the encoded X operator X_(L) at    level 2.    S _(Z){right arrow over (m _(z))}=0  (21-1)    S _(X){right arrow over (m _(x))}=0  (21-2)    Z _(L){right arrow over (m _(z))}=m _(z)  (21-3)    X _(L){right arrow over (m _(x))}=m _(x)  (21-4)

In this case, ((22-1)) and ((22-2)) hold.

$\begin{matrix}{\overset{arrow}{m_{z}} = \begin{pmatrix}m_{z}^{(1)} \\\vdots \\m_{z}^{(n)}\end{pmatrix}} & ( {22\text{-}1} ) \\{\overset{arrow}{m_{x}} = \begin{pmatrix}m_{x}^{(1)} \\\vdots \\m_{x}^{(n)}\end{pmatrix}} & ( {22\text{-}2} )\end{matrix}$

((20-1)) is executed by moving (n−1) variables of 2n variables of((23-1)) as independent variables, and determining (n+1) remainingvariables in accordance with above-mentioned conditions 5 to 8.

Subroutine 1 described above will be explained below with reference toFIG. 6. FIG. 6 is a flowchart of subroutine 1 of the probabilitycalculation unit of the decoding apparatus according to the firstembodiment when using a concatenated code obtained by concatenating Z·Xseparation type stabilizer codes.

Initial values R_(b2) (m_(z), m_(x)) (i=0) for taking the sum of theprobabilities ((19-1)) are set to zero (step S601). i takes 2^(n−1)values equal to the number of patterns of variables which ((23-1)) canindependently take.

After steps S402 and S403, of all components m_(z) ^((d)) and m_(x)^((d)) (to be also referred to as m_(z)[d] and m_(x)[d] hereinafter)(d=1, . . . , n) of two vectors contained in ((23-1)), each bit of i issubstituted into (n−1) components (step S604).

Of all the components m_(z) ^((d)) and m_(x) ^((d)), (n+1) remainingcomponents are determined in accordance with above-mentioned conditions5 to 8 (step S605).

P₀ is set to 1 (step S606).

Since d can take n values from 1 to n, steps S607 to S610 are repeatedby using each value of d.

In step S608, multiplication is performed by calculating P⁽¹⁾_(n(b2−1)+d)(m_(z) ^((d)),m_(x) ^((d))) on the right-hand side of (19-1)(step S608).

The values obtained in step S608 are added (step S609).

The calculation is advanced based on the value calculated for i lasttime, thereby finally obtaining the right-hand side of equation (19-1)above.

Steps S402, S403, and S604 to S610 are repeated immediately before iexceeds 2^(n−1).({right arrow over (m _(z))},{right arrow over (m _(x))})  (23-1)

Then, ((24-2)) is obtained by normalizing the relative probability((19-2)) as indicated by the following equation.

$\begin{matrix}{{P_{b_{2}}^{(2)}( {m_{z},m_{x}} )} = \frac{R_{b_{2}}^{(2)}( {m_{z},m_{x}} )}{\sum\limits_{({m_{z}^{\prime},m_{x}^{\prime}})}{R_{b_{2}}^{(2)}( {m_{z}^{\prime},m_{x}^{\prime}} )}}} & ( {24\text{-}1} ) \\{P_{b_{2}}^{(2)}( {m_{z},m_{x}} )} & ( {24\text{-}2} )\end{matrix}$

Subroutine 2 above will be explained below with reference to FIG. 7.FIG. 7 is a flowchart of subroutine 2 of the probability calculationunit of the decoding apparatus according to the first embodiment whenusing a concatenated code obtained by concatenating Z·X separation typestability codes.

Since each of m_(z) and m_(x) takes a value of 0 or 1, summationperformed by the denominator of the right-hand side of equation (24-1)can take four patterns. Steps S501, S702, and S503 are repeated forthese four patterns.

In step S702, ((24-2)) is obtained by performing the subroutine shown inFIG. 6 on the values of m_(z) and m_(x).

After the loop of steps S501, S702, and S503 is complete, the right-handside of equation (24-1) is calculated by ((24-2)) calculated by thisloop, thereby obtaining ((24-2)).

By using subroutine 2 described above, the probabilities ((25-1)) forthe eigenvalues m_(z) of the encoded Z operator at level 3 and theeigenvalues m_(x) of the encoded X operator at level 3 can be calculatedby using ((24-2)) with respect to the first qubit in the Bell state andeach block b₃ (b₃=1, 2, . . . , n^(L−3)) at level 3 in the input state.(Note that ((17-1)) shown in FIG. 6 is replaced with ((24-2)).) ((16-1))can be calculated by continuing the same calculation to level L.P _(b) ₃ ⁽³⁾(m _(z) ,m _(x))  (25-1)

The above-described operation of the probability calculation unit 204will be explained below with reference to FIG. 8. FIG. 8 is a flowchartshowing the operation of the probability calculation unit of thedecoding apparatus according to the first embodiment when using aconcatenated code obtained by concatenating Z·X separation typestabilizer codes.

In accordance with the subroutine shown in FIG. 5, the probabilities((17-1)) for the eigenvalues m_(z) of the encoded Z operator at level 1and the eigenvalues m_(x) of the encoded X operator at level 1 arecalculated with respect to the first qubit in the Bell state and eachblock b₁ at level 1 in the input state by using the above-describedcalculation method shown in FIGS. 4 and 5 (step S801).

The probabilities from level 1 to level L are calculated in the samemanner as when the probabilities at level 2 ((24-2)) are calculated fromthe probabilities at level 1 ((17-1)) in FIG. 7. After the probabilitiesof level k are calculated by the subroutine shown in FIG. 7, k is movedfrom 2 to L, and calculations are performed by a loop from step S802 tostep S804.

In the first embodiment described above, the measurement results of twoencoded qubits are simultaneously processed, and a measurement result isestimated and determined by soft-decision decoding based onprobabilistic inference. When compared to the conventional methods,therefore, it is possible to remarkably improve the decoding performanceof encoded Bell measurement used in error-correcting teleportation or inan encoded controlled-NOT gate. Consequently, the performance offault-tolerant quantum computation can be improved.

(Second Embodiment)

Next, a decoding apparatus according to the second embodiment will beexplained. FIG. 9 shows the decoding apparatus of this embodiment, andthe apparatus includes an M_(z) measurement value input unit 201, M_(x)measurement value input unit 202, error probability holding unit 203,probability calculation unit 204, and decision unit 901. Note that thesame reference numbers as in FIG. 2 denote components having the samefunctions in FIG. 9, a detailed explanation thereof will be omitted, anddifferent parts will mainly be explained below.

The decoding apparatus of the second embodiment can perform not onlyerror correction but also error detection, and this feature is obtainedby the decision unit 901.

The decision unit 901 first detects a maximum one of probabilities inputfrom the probability calculation unit 204. Then, the decision unit 901compares the detected maximum probability with a probability p_(det) (tobe called “an error detection determination probability” hereinafter)that is held in advance or externally input and held. If the maximumvalue of the probabilities is higher than the error detectiondetermination probability p_(det), the decision unit 901 outputscorresponding measurement values, and terminates the decoding process.On the other hand, if the maximum value of the probabilities is nothigher than the error detection determination probability p_(det), thedecision unit 901 decides that an error is detected, outputs a valuenotifying this, and terminates the decoding process.

Note that if the set value of the error detection determinationprobability p_(det) is too low, the performance may be inferior to thatof error detection using hard-decision decoding, so it is necessary tocarefully set p_(det). (See examples to be described later.)

The second embodiment described above can achieve the same effect asthat of the first embodiment, and can also perform error detection bycomparing the detected maximum probability with the error detectiondetermination probability.

(Third Embodiment)

A decoding apparatus according to the third embodiment will be explainedbelow. FIG. 10 shows the arrangement of the decoding apparatus, and theapparatus includes an M_(z) measurement value input unit 1001, M_(x)measurement value input unit 1002, error probability holding unit 1003,probability calculation unit 1004, and decision unit 205. Note that thesame reference numbers as in FIG. 2 denote components having the samefunctions in FIG. 10, a detailed explanation thereof will be omitted,and different parts will mainly be explained below.

The decoding apparatus of the third embodiment is a decoding apparatuscapable of performing a decoding process when errors include an erasureerror and probabilistic gate error. This feature is obtained by theM_(z) measurement value input unit 1001, M_(x) measurement value inputunit 1002, error probability holding unit 1003, and probabilitycalculation unit 1004.

The erasure error is an error by which whether there is an error in eachphysical qubit is known beforehand. On the other hand, the probabilisticgate error is an error caused by a gate (probabilistic gate) by whichwhether the gate is successful or unsuccessful is known beforehand whenexecuting the gate. If the gate is unsuccessful, the probabilistic errorcan be processed in the same manner as that for the erasure error byregarding that there is an erasure error in the physical qubit on whichthe gate is executed. Accordingly, only the erasure error will beexplained below.

The M_(z) measurement value input unit 1001 and M_(x) measurement valueinput unit 1002 respectively receive the measurement values of M_(z) andM_(x). If there is an erasure error, however, a value notifying theerror is input.

The M_(z) measurement value input unit 1001 and M_(x) measurement valueinput unit 1002 output these values not only to the probabilitycalculation unit 1004 but also to the error probability holding unit1003.

The error probability holding unit 1003 updates the error probabilitiesbased on the erasure error information input from the M_(z) measurementvalue input unit 1001 and M_(x) measurement value input unit 1002, andoutputs the updated error probabilities to the probability calculationunit 1004.

A standard method of updating the error probabilities is a method ofreplacing erased qubits with the average. Assume that two qubits exist,and P(e_(z1), e_(z2), e_(x1), e_(x2)) is prestored as the errorprobability. If a first bit is erased, the error probability is updatedas follows.

$\begin{matrix}{{P^{\prime}( {0,e_{z\; 2},0,e_{x\; 2}} )} = {P^{\prime}( {0,e_{z\; 2},1,e_{x\; 2}} )}} \\{= {P^{\prime}( {1,e_{z\; 2},0,e_{x\; 2}} )}} \\{= {P^{\prime}( {1,e_{z\; 2},1,e_{x\; 2}} )}} \\{= \frac{\begin{matrix}( {{P( {0,e_{z\; 2},0,e_{x\; 2}} )} + {P( {0,{e_{{z\; 2},}1},e_{x\; 2}} )} +}  \\ {{P( {1,e_{z\; 2},0,e_{x\; 2}} )} + {P( {1,e_{z\; 2},1,e_{x\; 2}} )}} )\end{matrix}}{4}}\end{matrix}$

The probability calculation unit 1004 calculates the probabilities forthe measurement values of Bell measurement, i.e., for the eigenvalues ofan encoded Z operator with respect to the first qubit in the Bell stateand the eigenvalues of an encoded X operator with respect to the inputstate, by using the measurement values and erasure error informationinput from the M_(z) measurement value input unit 1001 and M_(x)measurement value input unit 1002, the error probabilities input fromthe error probability holding unit 1003, and quantum error correctioncode information that is held in advance or externally input and held,and outputs the calculated probabilities to the decision unit 205.

The operation of the probability calculation unit 1004 is almost thesame as that of the probability calculation unit 204 of theabove-described first embodiment. If there is an erasure error, however,there is no measurement value ((26-1)) for a bit having the erasureerror (((26-2)) and ((26-3)) have a correlation in a physicalcontrolled-NOT gate, so the erasure error is processed by this bitpair), so the calculation of ((26-4)) seems to be impossible.

$\begin{matrix}( {m_{zj},m_{xj}} ) & ( {26\text{-}1} ) \\m_{zj} & ( {26\text{-}2} ) \\m_{xj} & ( {26\text{-}3} ) \\{{R( {m_{z},m_{x}} )} = {\sum\limits_{({\overset{arrow}{l_{z}},\overset{arrow}{l_{x}}})}{P^{(0)}( {{\overset{arrow}{l_{z}} + \overset{arrow}{m_{z}}},{\overset{arrow}{l_{x}} + \overset{arrow}{m_{x}}}} )}}} & ( {26\text{-}4} )\end{matrix}$When using the above-described standard method of updating the errorprobabilities, however, ((27-1)) is independent of the values of((26-1)), so ((26-1)) need only be calculated as an appropriate value(for example, ((27-2))).P ⁽⁰⁾({right arrow over (l _(z))}+{right arrow over (m _(z))},{rightarrow over (l _(x))}+{right arrow over (m _(x))})  (27-1)(m _(zj) ,m _(xj))=(0,0)  (27-2)

FIG. 11 shows results indicating that the performance of the abovedecoding method is higher than that of the conventional hard-decisiondecoding method. Referring to FIG. 11, thresholds for a communicationchannel having both an erasure error and normal undetectable error arecalculated by simulation. A Knill's C₄/C₆ code was used as the code (seeexamples). Although the C₄/C₆ code has two encoded qubits, one of themis used in this embodiment. The sum of encoded qubits not used inprobability calculations is calculated, and the probability for encodedqubits used in probability calculations is calculated (this applies tothe following examples). The probability at which a normal undetectableerror occurs under the condition that no erasure error has occurred willbe called a “conditional error probability”. Each plot in the graph ofFIG. 11 represents the threshold of this conditional error probabilitywith respect to a given erasure error probability (i.e., the upper limitof the conditional error probability at which the decoding errorprobability decreases when the level of a concatenated code isincreased). The solid circles indicate a result obtained when using thesoft-decision decoding method of this embodiment, and the solid squaresindicate a result obtained when using the conventional hard-decisiondecoding method. FIG. 11 reveals that the method of this embodiment hasperformance much higher than that of the conventional method. Thethreshold is about 19% when the erasure error probability is zero, andthis value matches the theoretical limit known as a hashing bound. Also,the threshold is zero when the erasure error probability is 50%, andthis value matches the theoretical limit of an erasure error.

The third embodiment described above can achieve the same effect as thatof the first embodiment. In addition, even when errors include anerasure error and probabilistic gate error, it is possible to update theerror probability based on erasure error information, and obtain themeasurement values of Bell measurement by using the standard method ofupdating the error probabilities.

(Fourth Embodiment)

A decoding apparatus according to the fourth embodiment will beexplained below. FIG. 12 shows the arrangement of the decodingapparatus, and the apparatus includes an M_(z) measurement value inputunit 1001, M_(x) measurement value input unit 1002, error probabilityholding unit 1003, probability calculation unit 1004, and decision unit901. Note that the same reference numbers as in FIGS. 9 and 10 denotecomponents having the same functions in FIG. 12, a detailed explanationthereof will be omitted, and different parts will mainly be explainedbelow.

The decoding apparatus of the fourth embodiment is a decoding apparatuscapable of performing not only error correction but also error detectionwhen errors include an erasure error and probabilistic gate error. Thisfeature is obtained by combining the M_(z) measurement value input unit1001, M_(x) measurement value input unit 1002, error probability holdingunit 1003, and probability calculation unit 1004 of the decodingapparatus of the third embodiment, and the decision unit 901 of thedecoding apparatus of the second embodiment.

The decision unit 901 first receives probabilities calculated by theprobability calculation unit 1004 by taking account of erasure errorinformation, and detects a maximum one of the probabilities. Then, thedecision unit 901 compares the maximum probability with an errordetection determination probability p_(det) that is held in advance orexternally input and held. If the maximum value of the probabilities ishigher than the error detection determination probability p_(det), thedecision unit 901 outputs corresponding measurement values, andterminates the decoding process. On the other hand, if the maximum valueof the probabilities is not higher than the error detectiondetermination probability p_(det), the decision unit 901 decides that anerror is detected, outputs a value notifying this, and terminates thedecoding process.

The decoding apparatus of the fourth embodiment is very useful in statepreparations when an erasure error or probabilistic gate error exists(see examples).

The fourth embodiment described above can simultaneously achieve theeffects of the second and third embodiments.

(Fifth Embodiment)

Next, an encoded controlled-NOT gate according to the fifth embodimentwill be explained. The decoding apparatus according to the embodiment isapplied to this encoded controlled-NOT gate. FIG. 13 shows thearrangement of the gate.

This encoded controlled-NOT gate using ((28-1)) is already known (seeHayato Goto and Koichi Ichimura, Japanese Patent No. 4786727, and H.Goto and K. Ichimura, Phys. Rev. A 80, 040303(R) [2009]). Theperformance of this encoded controlled-NOT gate can dramatically beimproved by using the decoding apparatus of the embodiment in encodedBell measurement performed by the gate (see examples).|χ

_(L)  (28-1)Note that an entangled state ((28-1)) including four encoded qubits isdefined by equation (29-1) below.|χ

_(L)=|0

_(L)|0

_(L)|0

_(L)|0

_(L)+|0

_(L)|0

_(L)|1

_(L)|1

_(L)+|1

_(L)|1

_(L)|0

_(L)|1

_(L)+|1

_(L)|1

_(L)|1

_(L)|0

_(L)  (29-1)

The fifth embodiment described above can dramatically improve theperformance of an encoded controlled-NOT gate by applying the decodingapparatus according to one of the first to fourth embodiments to theencoded controlled-NOT gate.

EXAMPLES

Examples will now be explained.

Since a C₄/C₆ code (see E. Knill, Nature 434, 39 [2005]) was used as aquantum error correction code in all examples below, the C₄/C₆ code isexplained first.

The C₄/C₆ code is a C₄ code at level 1, and is a concatenated code usinga C₆ code at level 2 or higher.

The C₄ code is a Z·X separation type stabilizer code that encodes twoqubits by four qubits. A Z stabilizer generator is ZZZZ, and an Xstabilizer generator is XXXX. Also, encoded Z gates are ZIZI and IIZZ,and encoded X gates are XXII and IXIX. (Two pairs exist in order toencode two qubits.)

The C₆ code is a Z·X separation type stabilizer code that encodes twoqubits by six qubits. Z stabilizer generators are ZIIZZZ and ZZZIIZ, andX stabilizer generators are XIIXXX and XXXIIX. Also, encoded Z gates areIIZZIZ and IIIZZI, and encoded X gates are IXXIII and XIXXII. (Two pairsexist in order to encode two qubits.)

As is apparent from the above description, the C₄/C₆ code is aconcatenated code obtained by concatenating Z·X separation typestabilizer codes, and the efficient probability calculation algorithmsshown in FIGS. 5 to 9 are applicable. In all examples below, acalculation program of a probability calculation unit is based on thealgorithms shown in FIGS. 5 to 9.

In addition, in all examples below, as the error probability of an errorprobability holding unit, the error probability ((30-2)) for an errore_(zj) for a measurement value m_(zj) and an error e_(xj) for ameasurement value m_(xj) is set as follows for all bit pairs j.

$\begin{matrix}{{P_{j}^{(0)}( {e_{zj},e_{xj}} )} = \{ \begin{matrix}{{0.81\mspace{14mu}\ldots\mspace{14mu} e_{zj}} = {e_{xj} = 0}} \\{{{0.19/3}\mspace{14mu}\ldots\mspace{14mu}{if}\mspace{14mu}{not}\mspace{14mu} e_{zj}} = {e_{xj} = 0}}\end{matrix} } & ( {30\text{-}1} ) \\{P_{j}^{(0)}( {e_{zj},e_{xj}} )} & ( {30\text{-}2} )\end{matrix}$The present inventors confirmed by simulation that even when a fixedvalue is thus set, the value is effective for actual various errorprobabilities. That is, the decoding method of the embodiment is robustfor the way of setting ((30-2)). This is an important feature because itis difficult to accurately estimate an error probability byfault-tolerant quantum computation. It should be noted that thecorrelation between e_(zj) and e_(xj) is taken into consideration.

In all examples below, encoded controlled-NOT gates are evaluated bycomputer simulation.

In the first and second examples processing normal error models,physical controlled-NOT gates are transversally executed in accordancewith E. Knill, Nature 434, 39 (2005), and two error-correctingteleportations are performed. The decoding apparatus of the embodimentis used in the error-correcting teleportations.

Also, in the third and fourth examples processing probabilistic gatemodels, encoded controlled-NOT gates using ((28-1)) are performed inaccordance with H. Goto and K. Ichimura, Phys. Rev. A 80, 040303(R)(2009). In these examples, the encoded controlled-NOT gates of the fifthembodiment are used.

The first and second examples used a state preparation method complyingwith E. Knill, Nature 434, 39 (2005), and the third and fourth examplesused a state preparation method complying with H. Goto and K. Ichimura,Phys. Rev. A 80, 040303(R) (2009).

In the above-described state preparations, error detection and postselection based on the error detection are performed. Since the methodchanged from one example to another, these methods will be explainedbelow.

First Example

In this example, to perform an encoded controlled-NOT gate, a physicalcontrolled-NOT gate is first transversally executed, and thenerror-correcting teleportation is executed on each of two encodedqubits. The decoding apparatus of the first embodiment is used in theseprocesses.

First, an error model used in this example will be explained. Assumingthat an error existed in only a physical controlled-NOT gate, a standardmodel called a depolarizing model is used (see E. Knill, Nature 434, 39[2005]). Let p_(e) be the error probability of the model.

In the state preparations, error detection and post selection based onthe error detection are performed. In this example, the error detectionin the state preparations is performed using only conventionalhard-decision decoding error correction (see E. Knill, Nature 434, 39[2005]).

FIG. 14 shows simulation results comparing the performance (the hollowcircles) when using the decoding apparatus (FIG. 3) of the firstembodiment with the performance (the x-marks) when using a conventionalhard-decision decoding apparatus. Note that p_(e) is set to 0.01.

As shown in FIG. 14, the error probability of the method of theembodiment is lower by two orders of magnitude than that of theconventional method at level 4. On the other hand, the resources (thenumbers of times of the physical controlled-NOT gate) are the same.

As described above, when using the decoding apparatus (FIG. 2) of thefirst embodiment instead of the conventional hard-decision decodingapparatus, the performance of an encoded controlled-NOT gate and hencethe performance of fault-tolerant quantum computation dramaticallyimproved.

Second Example

In this example, the decoding apparatus (FIG. 9) of the secondembodiment is applied to error detection in the state preparations atlevel 4 in the first example. (Until level 3, the performance ofhard-decision decoding error detection is almost the same as that ofoptimal soft-decision decoding, so hard-decision decoding requiring asmall calculation amount is used.) An error model used in this exampleis the same as that used in the first example.

The error detection determination probability p_(det) is set to 0.997.

The state preparations require not only error detection in encoded Bellmeasurement, but also decoding and error detection on only themeasurement value of M_(z) and decoding and error detection on only themeasurement value of M_(x). In this example, soft-decision decoding wasused in these processes as well. A decoding apparatus for only the Mameasurement value is obtained by changing the decoding apparatus of thesecond embodiment such that the M_(z) measurement value input unit 201alone is used as an input unit, the error probability is changed to anerror probability ((31-1)) for the measurement value m_(zj), andvariables to be used in the calculation of the probability calculationunit are changed to only variables concerning Z. This similarly appliesto a decoding apparatus for only the M_(x) measurement value.P _(j) ⁽⁰⁾(e _(zj))  (31-1)An error detection determination probability p′_(det) of the decodingapparatus for only the M_(z) measurement value and the decodingapparatus for only the M_(x) measurement value is set to 0.995.

When p_(e)=1%, the error probability of the encoded controlled-NOT gate(at level 4) of this example is about 0.0002%. This is half or less anerror probability of 0.00045% (see FIG. 14) of the encodedcontrolled-NOT gate of the first example. On the other hand, theresources (the numbers of times of the physical controlled-NOT gate) arealmost the same.

As described above, when using the decoding apparatus of the secondembodiment shown in FIG. 9 in error detection performed in the statepreparations, the performance of the encoded controlled-NOT gate andhence the performance of fault-tolerant quantum computation furtherimproved.

Third Example

In this example, the decoding apparatus of the third embodiment isapplied to the encoded controlled-NOT gate of the fifth embodiment.

First, an error model used in this example will be explained. Assumingthat an error exists in only a physical controlled-NOT gate, and aprobabilistic gate model is used as the model (see Hayato Goto andKoichi Ichimura, Japanese Patent No. 4786727, and H. Goto and K.Ichimura, Phys. Rev. A 80, 040303(R) [2009]). In this error model,whether the physical controlled-NOT gate is successful or unsuccessfulis known when it is executed. Let p_(F) be this failure probability. Ifthe physical controlled-NOT gate is unsuccessful, it is determined thattwo physical qubits on which the gate has been executed have an error,and the error is regarded as an erasure error. If the physicalcontrolled-NOT gate is successful, a depolarizing error occurs at aconditional error probability p_(c).

In this example, error detection in the state preparations is performedusing only conventional hard-decision decoding error detection (seeHayato Goto and Koichi Ichimura, Japanese Patent No. 4786727, and H.Goto and K. Ichimura, Phys. Rev. A 80, 040303(R) [2009]).

Simulation is performed under the conditions that p_(F)=0.05 andp_(c)=0.004. FIG. 15 shows the results. The hollow circles indicate theresult when using the decoding apparatus (FIG. 10) of the thirdembodiment. The x-marks indicate the result when using the conventionalhard-decision decoding apparatus.

At levels 3 and 4, the error probability of the method of the thirdembodiment is about ⅓ that of the conventional method. On the otherhand, the resources (the numbers of times of the physical controlled-NOTgate) are the same.

As described above, since the decoding apparatus (FIG. 10) of the thirdembodiment is used instead of the conventional hard-decision decodingapparatus, the performance of the encoded controlled-NOT gate when usingthe probabilistic gate and hence the performance of fault-tolerantquantum computation when using the probabilistic gate dramaticallyimproved.

Fourth Example

In this example, the decoding apparatus (FIG. 12) of the fourthembodiment is applied to error detection in the state preparations atlevels 3 and 4 in the third example.

An error model used in this example is the same as that of the thirdexample.

An error detection determination probability p_(det) is set to 0.99 atlevel 3, and to 0.997 at level 4. Also, an error detection determinationprobability p′_(det) of a decoding apparatus for M_(z) measurementvalues alone and a decoding apparatus for M_(x) measurement values aloneis set to 0.9 at level 3, and to 0.97 at level 4.

Simulation was performed under the conditions that p_(F)=0.05 andp_(C)=0.004. FIG. 16 shows the results. The hollow squares indicate theresult of this example, and the hollow circles indicate the result ofthe first example. At level 4, the error probability of this example islower by nearly one order of magnitude than that of the third example.On the other hand, the resources (the numbers of times of the physicalcontrolled-NOT gate) are almost the same.

As described above, since the decoding apparatus (FIG. 12) of the fourthembodiment is used in error detection performed in the statepreparations, the performance of the encoded controlled-NOT gate whenusing the probabilistic gate and hence the performance of fault-tolerantquantum computation when using the probabilistic gate further improved.

The flow charts of the embodiments illustrate methods and systemsaccording to the embodiments. It will be understood that each block ofthe flowchart illustrations, and combinations of blocks in the flowchartillustrations, can be implemented by computer program instructions.These computer program instructions may be loaded onto a computer orother programmable apparatus to produce a machine, such that theinstructions which execute on the computer or other programmableapparatus create means for implementing the functions specified in theflowchart block or blocks. These computer program instructions may alsobe stored in a computer-readable memory that can direct a computer orother programmable apparatus to function in a particular manner, suchthat the instruction stored in the computer-readable memory produce anarticle of manufacture including instruction means which implement thefunction specified in the flowchart block or blocks. The computerprogram instructions may also be loaded onto a computer or otherprogrammable apparatus to cause a series of operational steps to beperformed on the computer or other programmable apparatus to produce acomputer programmable apparatus which provides steps for implementingthe functions specified in the flowchart block or blocks.

While certain embodiments have been described, these embodiments havebeen presented by way of example only, and are not intended to limit thescope of the inventions. Indeed, the novel embodiments described hereinmay be embodied in a variety of other forms; furthermore, variousomissions, substitutions and changes in the form of the embodimentsdescribed herein may be made without departing from the spirit of theinventions. The accompanying claims and their equivalents are intendedto cover such forms or modifications as would fall within the scope andspirit of the inventions.

What is claimed is:
 1. A decoding apparatus for use in encoded Bellmeasurement for two encoded qubits, comprising: a first acquisition unitconfigured to acquire first measurement values of measurements performedto measure an eigenvalue of an encoded Z operator with respect to afirst encoded qubit of the two encoded qubits; a second acquisition unitconfigured to acquire second measurement values of measurementsperformed to measure an eigenvalue of an encoded X operator with respectto a second encoded qubit of the two encoded qubits; a holding unitconfigured to hold error probabilities for the first measurement valuesand the second measurement values; a calculation unit configured tocalculate probabilities for measurement values of the encoded Bellmeasurement by using the first measurement values, the secondmeasurement values, and the error probabilities; and a decision unitconfigured to decide measurement values of the encoded Bell measurement,based on the probabilities calculated by the calculation unit, whereinthe eigenvalue of the encoded Z operator and the eigenvalue of theencoded X operator being measurement values of the Bell measurement. 2.The apparatus according to claim 1, wherein the decision unit detects amaximum probability among the probabilities calculated by thecalculation unit, and decides that a measurement result corresponding tothe maximum probability is the measurement result of the encoded Bellmeasurement.
 3. The apparatus according to claim 1, wherein the decisionunit detects a maximum probability among the probabilities calculated bythe calculation unit, compares the maximum probability with an errordetection determination probability held in advance, decides thatmeasurement values corresponding to the maximum probability are themeasurement values of the encoded Bell measurement if the maximumprobability is higher than the error detection determinationprobability, and decides that an error is detected if the maximumprobability is not higher than the error detection determinationprobability.
 4. The apparatus according to claim 1, wherein the firstacquisition unit further acquires erasure information concerningphysical qubits forming the first encoded qubit, and probabilistic gatefailure information, the second acquisition unit further acquireserasure information concerning physical qubits forming the secondencoded qubit, and probabilistic gate failure information, the holdingunit updates the error probability based on the erasure information andthe probabilistic gate failure information acquired by the firstacquisition unit, and the erasure information and the probabilistic gatefailure information acquired by the second acquisition unit, and thecalculation unit calculates the probabilities for the measurement valuesof the encoded Bell measurement by using the erasure information and theprobabilistic gate failure information acquired by the first acquisitionunit, the erasure information and the probabilistic gate failureinformation acquired by the second acquisition unit, and the errorprobabilities updated by the holding unit.
 5. A decoding apparatus foruse in encoded Bell measurement for two encoded qubits encoded by usinga Z·X separation type stabilizer code, comprising: a first acquisitionunit configured to acquire first measurement values of Z operatoreigenvalue measurements performed for physical qubits forming a firstencoded qubit of the two encoded qubits, in order to measure aneigenvalue of an encoded Z operator for the first encoded qubit; asecond acquisition unit configured to acquire second measurement valuesof X operator eigenvalue measurements performed for physical qubitsforming a second encoded qubit of the two encoded qubits, in order tomeasure an eigenvalue of an encoded X operator for the second encodedqubit; a holding unit configured to hold error probabilities for thefirst measurement values and the second measurement values; acalculation unit configured to calculate probabilities for measurementvalues of the encoded Bell measurement by using the first measurementvalues, the second measurement values, and the error probabilities; anda decision unit configured to decide measurement values of the encodedBell measurement, based on the probabilities calculated by thecalculation unit wherein the eigenvalue of the encoded Z operator andthe eigenvalue of the encoded X operator being measurement values of theBell measurement.
 6. The apparatus according to claim 5, wherein the Z·Xseparation type stabilizer code is a concatenated code obtained byconcatenating Z·X separation type stabilizer codes.
 7. The apparatusaccording to claim 6, wherein the calculation unit calculatesprobabilities for measurement values of each block at level 1 of theconcatenated code by using the first measurement values, the secondmeasurement values, the error probabilities, and level 1 encodinginformation of the concatenated code, and calculates probabilities formeasurement values of each block at level (k+1) (k is an integer of notless than 1) of the concatenated code, by using probabilities formeasurement values of each block at level k of the concatenated code,and level (k+1) encoding information of the concatenated code.
 8. Theapparatus according to claim 5, wherein the decision unit detects amaximum probability among the probabilities calculated by thecalculation unit, and decides that a measurement result corresponding tothe maximum probability is the measurement result of the encoded Bellmeasurement.
 9. The apparatus according to claim 5, wherein the decisionunit detects a maximum probability among the probabilities calculated bythe calculation unit, compares the maximum probability with an errordetection determination probability held in advance, decides thatmeasurement values corresponding to the maximum probability are themeasurement values of the encoded Bell measurement if the maximumprobability is higher than the error detection determinationprobability, and decides that an error is detected if the maximumprobability is not higher than the error detection determinationprobability.
 10. The apparatus according to claim 5, wherein the firstacquisition unit further acquires erasure information concerningphysical qubits forming the first encoded qubit, and probabilistic gatefailure information, the second acquisition unit further acquireserasure information concerning physical qubits forming the secondencoded qubit, and probabilistic gate failure information, the holdingunit updates the error probabilities based on the erasure informationand the probabilistic gate failure information acquired by the firstacquisition unit, and the erase information and the probabilistic gatefailure information acquired by the second acquisition unit, and thecalculation unit calculates the probabilities for the measurement valuesof the encoded Bell measurement by using the erasure information and theprobabilistic gate failure information acquired by the first acquisitionunit, the erasure information and the probabilistic gate failureinformation acquired by the second acquisition unit, and the errorprobabilities updated by the holding unit.
 11. A decoding method for usein encoded Bell measurement for two encoded qubits, comprising:acquiring first measurement values of measurements performed to measurean eigenvalue of an encoded Z operator with respect to a first encodedqubit of the two encoded qubits; acquiring second measurement values ofmeasurements performed to measure an eigenvalue of an encoded X operatorwith respect to a second encoded qubit of the two encoded qubits;calculating probabilities for measurement values of the encoded Bellmeasurement by using the first measurement values, the secondmeasurement values, and an error probabilities for the first measurementvalues and the second measurement values; and deciding measurementvalues of the encoded Bell measurement, based on the calculatedprobabilities, wherein the eigenvalue of the encoded Z operator and theeigenvalue of the encoded X operator being measurement values of theBell measurement.
 12. A decoding method for use in encoded Bellmeasurement for two encoded qubits encoded by using a Z·X separationtype stabilizer code, comprising: acquiring first measurement values ofZ operator eigenvalue measurements performed for physical qubits forminga first encoded qubit of the two encoded qubits, in order to measure aneigenvalue of an encoded Z operator for the first encoded qubit;acquiring second measurement values of X operator eigenvaluemeasurements performed for physical qubits forming a second encodedqubit of the two encoded qubits, in order to measure an eigenvalue of anencoded X operator for the second encoded qubit; calculatingprobabilities for measurement values of the encoded Bell measurement byusing the first measurement values, the second measurement values, anderror probabilities for the first measurement values and the secondmeasurement values; and deciding measurement values of the encoded Bellmeasurement, based on the calculated probabilities, wherein theeigenvalue of the encoded Z operator and the eigenvalue of the encoded Xoperator being measurement values of the Bell measurement.